Following Cheeger and Gromov, I show that Betti numbers vanish for infinite amenable groups. I make the simplifying assumption that group acts freely and cocompactly on a contractible simplicial complex .

The result obviously follows from

Lemma 1 The forgetful map to ordinary cohomology

is injective.

We study the -dimension of the kernel of the forgetful map. By definition,

Let be a fundamental domain. Let be an increasing F\o lner sequence of finite subsets of . Let . Then

The -dimension can be rewritten

Let denote restriction of cochains to . I claim that

Indeed, the composition is a contraction, so . Thus the block of the matrix of , , satisfies

Now I claim that

Finally, for ,

If the support of does not meet , then , and thus and . Thus contains all the kernels of linear froms “evaluation on simplices of ”. This yield the codimension estimate.